Find the first 4 terms of the Laurent series centered at z0=0 for 1/ ( e^x - 1)
using division of power series.
Then, give a geometric reason why this expansion is valid on the puctured disk 0 < [z] <2pi

where [z] stands for absolute value of z.

Thank you .

Apr 24th 2010, 03:38 PM

mr fantastic

Quote:

Originally Posted by altanl

Find the first 4 terms of the Laurent series centered at z0=0 for 1/ ( e^z - 1)
using division of power series.
Then, give a geometric reason why this expansion is valid on the puctured disk 0 < [z] <2pi

where [z] stands for absolute value of z.

Thank you .

Correction in red.

The first step is to substitute the power series for e^z and simplify. Then divide. Have you done this? Where are you stuck? What have you tried?

By the way, after doing the first step it should be obvious that z = 0 is a simple pole. Therefore the series has the form .